Math 447: Intro to Partial Differential Equations

Title

Introduction to Partial Differential Equations.

(3:3:0)

W, Su

Prerequisite

Math 303; or 314 and 334.

Description

Boundary value problems; transform methods; Fourier series; Bessel functions; Legendre polynomials.

Desired Learning Outcomes

Minimal learning outcomes

The focus of the course is on helping students solve the canonical linear second-order partial differential equations on simple domains.

1. Basic classification of PDEs
• As nonlinear, linear homogeneous, linear inhomogeneous
• By order
• Of second-order linear PDEs in 2 variables as elliptic, parabolic, or hyperbolic
2. Basic Modeling
• Derivation of the heat equation
• Derivation of the wave equation
• Derivation of Dirichlet, Neumann, and mixed boundary conditions for the heat equation
3. Basic principles, techniques, and theory
• Principle of superposition
• Method of separation of variables
• Definition of eigenvalues and eigenfunctions corresponding to two-point BVPs
• Basic Sturm-Liouville theory
4. Special eigensystems
• Fourier series
• Computation of the Fourier series of a p-periodic function on an interval of length p
• Computation of Fourier sine and cosine series of a symmetric p-periodic function on an interval of length p
• Fourier series of modifications and combinations of functions
• Theorems on pointwise, uniform, and L2 convergence
• Bessel's Inequality and Parseval's Equation
• Fourier integral representations of functions on lines and half-lines
• Bessel's equation and Bessel functions
• Legendre's differential equation and Legendre polynomials
5. Representation of solutions to the canonical equations on simple domains
• Laplace's equation
• Eigenfunction expansion of solutions on rectangles
• Integral representation of solutions on rectangular strips, quarter-planes, and half-planes
• Bessel function expansion of solutions on disks
• Legendre polynomial expansion on balls
• Wave equation
• Fourier expansion of solutions on bounded intervals
• D'Alembert's formula for solutions on lines and half-lines
• Bessel function expansion of solutions on disks
• Legendre polynomial expansion on balls
• Heat equation
• Steady-state solutions for IBVPs on bounded intervals
• Fourier expansion of solutions on bounded intervals
• Integral representation of solutions on lines and half-lines
• Eigenfunction expansion of solutions on rectangles
• Bessel function expansion of solutions on disks
• Legendre polynomial expansion of solutions on balls